cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A194446 Number of parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

1, 2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 2, 1, 15, 1, 2, 1, 4, 1, 1, 22, 1, 2, 1, 4, 1, 2, 1, 30, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 42, 1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 3, 1, 1, 56, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 1, 77, 1, 2, 1
Offset: 1

Views

Author

Omar E. Pol, Nov 26 2011

Keywords

Comments

For the definition of "region" of the set of partitions of j, see A206437.
a(n) is also the number of positive integers in the n-th row of triangle A186114. a(n) is also the number of positive integers in the n-th row of triangle A193870.
Also triangle read by rows: T(j,k) = number of parts in the k-th region of the last section of the set of partitions of j. See example. For more information see A135010.
a(n) is also the length of the n-th vertical line segment in the minimalist diagram of regions and partitions. The length of the n-th horizontal line segment is A141285(n). See also A194447. - Omar E. Pol, Mar 04 2012
From Omar E. Pol, Aug 19 2013: (Start)
In order to construct this sequence with a cellular automaton we use the following rules: We start in the first quadrant of the square grid with no toothpicks. At stage n we place A141285(n) toothpicks of length 1 connected by their endpoints in horizontal direction starting from the point (0, n). Then we place toothpicks of length 1 connected by their endpoints in vertical direction starting from the exposed toothpick endpoint downward up to touch the structure or up to touch the x-axis. a(n) is the number of toothpicks in vertical direction added at n-th stage (see example section and A139250, A225600, A225610).
a(n) is also the length of the n-th descendent line segment in an infinite Dyck path in which the length of the n-th ascendent line segment is A141285(n). See Example section. For more information see A211978, A220517, A225600.
(End)
The equivalent sequence for compositions is A006519. - Omar E. Pol, Aug 22 2013

Examples

			Written as an irregular triangle the sequence begins:
  1;
  2;
  3;
  1, 5;
  1, 7;
  1, 2, 1, 11;
  1, 2, 1, 15;
  1, 2, 1,  4, 1, 1, 22;
  1, 2, 1,  4, 1, 2,  1, 30;
  1, 2, 1,  4, 1, 1,  7,  1, 2, 1, 1, 42;
  1, 2, 1,  4, 1, 2,  1,  8, 1, 1, 3,  1, 1, 56;
  1, 2, 1,  4, 1, 1,  7,  1, 2, 1, 1, 12, 1,  2, 1, 4, 1, 2, 1, 1, 77;
  ...
From _Omar E. Pol_, Aug 18 2013: (Start)
Illustration of initial terms (first seven regions):
.                                             _ _ _ _ _
.                                     _ _ _  |_ _ _ _ _|
.                           _ _ _ _  |_ _ _|       |_ _|
.                     _ _  |_ _ _ _|                 |_|
.             _ _ _  |_ _|     |_ _|                 |_|
.       _ _  |_ _ _|             |_|                 |_|
.   _  |_ _|     |_|             |_|                 |_|
.  |_|   |_|     |_|             |_|                 |_|
.
.   1     2       3     1         5       1           7
.
The next figure shows a minimalist diagram of the first seven regions. The n-th horizontal line segment has length A141285(n). a(n) is the length of the n-th vertical line segment, which is the vertical line segment ending in row n (see also A225610).
.      _ _ _ _ _
.  7   _ _ _    |
.  6   _ _ _|_  |
.  5   _ _    | |
.  4   _ _|_  | |
.  3   _ _  | | |
.  2   _  | | | |
.  1    | | | | |
.
.      1 2 3 4 5
.
Illustration of initial terms from an infinite Dyck path in which the length of the n-th ascendent line segment is A141285(n). a(n) is the length of the n-th descendent line segment.
.                                    /\
.                                   /  \
.                      /\          /    \
.                     /  \        /      \
.            /\      /    \    /\/        \
.       /\  /  \  /\/      \  / 1          \
.    /\/  \/    \/ 1        \/              \
.     1   2     3           5               7
.
(End)

Links

Crossrefs

Row j has length A187219(j). Right border gives A000041, j >= 1. Records give A000041, j >= 1. Row sums give A138137.
Cf. A002865, A006128, A135010, A138121, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194447.

Programs

  • Mathematica
    lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
A194446 = {}; l = {};
For[j = 1, j <= 30, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  AppendTo[A194446, j - i];
  ];
A194446   (* Robert Price, Jul 25 2020 *)

Formula

a(n) = A141285(n) - A194447(n). - Omar E. Pol, Mar 04 2012

A194447 Rank of the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

0, 0, 0, 1, -1, 2, -2, 1, 2, 2, -5, 2, 3, 3, -8, 1, 2, 2, 2, 4, 3, -14, 2, 3, 3, 3, 2, 4, 4, -21, 1, 2, 2, 2, 4, 3, 1, 3, 5, 5, 4, -32, 2, 3, 3, 3, 2, 4, 4, 1, 4, 3, 5, 6, 5, -45, 1, 2, 2, 2, 4, 3, 1, 3, 5, 5, 4, -2, 2, 4, 4, 5, 3, 6, 6, 5, -65
Offset: 1

Views

Author

Omar E. Pol, Dec 04 2011

Keywords

Comments

Here the rank of a "region" is defined to be the largest part minus the number of parts (the same idea as the Dyson's rank of a partition).
Also triangle read by rows: T(j,k) = rank of the k-th region of the last section of the set of partitions of j.
The sum of every row is equal to zero.
Note that in some rows there are several negative terms. - Omar E. Pol, Oct 27 2012
For the definition of "region" see A206437. See also A225600 and A225610. - Omar E. Pol, Aug 12 2013

Examples

			In the triangle T(j,k) for j = 6 the number of regions in the last section of the set of partitions of 6 is equal to 4. The first region given by [2] has rank 2-1 = 1. The second region given by [4,2] has rank 4-2 = 2. The third region given by [3] has rank 3-1 = 2. The fourth region given by [6,3,2,2,1,1,1,1,1,1,1] has rank 6-11 = -5 (see below):
From _Omar E. Pol_, Aug 12 2013: (Start)
---------------------------------------------------------
.    Regions       Illustration of ranks of the regions
---------------------------------------------------------
.    For J=6        k=1     k=2      k=3        k=4
.  _ _ _ _ _ _                              _ _ _ _ _ _
. |_ _ _      |                     _ _ _   .          |
. |_ _ _|_    |           _ _ _ _   * * .|    .        |
. |_ _    |   |     _ _   * * .  |              .      |
. |_ _|_ _|_  |     * .|        .|                .    |
.           | |                                     .  |
.           | |                                       .|
.           | |                                       *|
.           | |                                       *|
.           | |                                       *|
.           | |                                       *|
.           |_|                                       *|
.
So row 6 lists:     1       2         2              -5
(End)
Written as a triangle begins:
0;
0;
0;
1,-1;
2,-2;
1,2,2,-5;
2,3,3,-8;
1,2,2,2,4,3,-14;
2,3,3,3,2,4,4,-21;
1,2,2,2,4,3,1,3,5,5,4,-32;
2,3,3,3,2,4,4,1,4,3,5,6,5,-45;
1,2,2,2,4,3,1,3,5,5,4,-2,2,4,4,5,3,6,6,5,-65;
2,3,3,3,2,4,4,1,4,3,5,6,5,-3,3,5,5,4,5,4,7,7,6,-88;

Crossrefs

Row j has length A187219(j). The absolute value of the last term of row j is A000094(j+1). Row sums give A000004.
Cf. A000041, A002865, A135010, A138121, A138137, A138879, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A206437.

Formula

a(n) = A141285(n) - A194446(n). - Omar E. Pol, Dec 05 2011

A186412 Sum of all parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 3, 7, 4, 25, 2, 6, 3, 13, 5, 4, 38, 3, 7, 4, 14, 3, 9, 5, 49, 2, 6, 3, 13, 5, 4, 23, 4, 10, 6, 5, 69, 3, 7, 4, 14, 3, 9, 5, 27, 5, 4, 15, 7, 6, 87, 2, 6, 3, 13, 5, 4, 23, 4, 10, 6, 5, 39, 3, 9, 5, 19, 4, 12, 7, 6, 123
Offset: 1

Views

Author

Omar E. Pol, Aug 12 2011

Keywords

Comments

Also triangle read by rows: T(j,k) = sum of all parts in the k-th region of the last section of the set of partitions of j. See Example section. For more information see A135010. - Omar E. Pol, Nov 26 2011
For the definition of "region" see A206437. - Omar E. Pol, Aug 19 2013

Examples

			Contribution from Omar E. Pol, Nov 26 2011 (Start):
Written as a triangle:
1;
3;
5;
2,9;
3,12;
2,6,3,20;
3,7,4,25;
2,6,3,13,5,4,38;
3,7,4,14,3,9,5,49;
2,6,3,13,5,4,23,4,10,6,5,69;
3,7,4,14,3,9,5,27,5,4,15,7,6,87;
2,6,3,13,5,4,23,4,10,6,5,39,3,9,5,19,4,12,7,6,123;
(End)
From _Omar E. Pol_, Aug 18 2013: (Start)
Illustration of initial terms (first seven regions):
.                                             _ _ _ _ _
.                                     _ _ _  |_ _ _ _ _|
.                           _ _ _ _  |_ _ _|       |_ _|
.                     _ _  |_ _ _ _|                 |_|
.             _ _ _  |_ _|     |_ _|                 |_|
.       _ _  |_ _ _|             |_|                 |_|
.   _  |_ _|     |_|             |_|                 |_|
.  |_|   |_|     |_|             |_|                 |_|
.
.   1     3       5     2         9       3          12
.
(End)

Crossrefs

Row sums of triangle A186114 and of triangle A193870.
Row j has length A187219(j).
Row sums give A138879.
Right border gives A046746, j >= 1.
Records give A046746, j >= 1.
Partial sums give A182244.
Cf. A000041, A002865, A066186, A135010, A138121, A138879, A194436, A194437, A194438, A194439, A194446, A194447.

Programs

  • Mathematica
    lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
A186412 = {}; l = {};
For[j = 1, j <= 50, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  AppendTo[A186412, Total@Take[Reverse[First /@ lex[mx]], j - i]];
  ];
A186412  (* Robert Price, Jul 25 2020 *)

Formula

a(A000041(n)) = A046746(n).

A194436 Triangle read by rows: T(n,k) = number of parts in the k-th region of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 1, 5, 1, 2, 3, 1, 5, 1, 7, 1, 2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 2, 1, 15, 1, 2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 2, 1, 15, 1, 2, 1, 4, 1, 1, 22, 1, 2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 2, 1, 15
Offset: 1

Views

Author

Omar E. Pol, Nov 27 2011

Keywords

Examples

			Triangle begins:
1;
1,2;
1,2,3;
1,2,3,1,5;
1,2,3,1,5,1,7;
1,2,3,1,5,1,7,1,2,1,11;
1,2,3,1,5,1,7,1,2,1,11,1,2,1,15;
1,2,3,1,5,1,7,1,2,1,11,1,2,1,15,1,2,1,4,1,1,22;
...
Row n has length A000041(n). Row sums give A006128, n >= 1. Right border gives A000041, n >= 1. Records in every row give A000041, n >= 1. Rows converge to A194446.

Crossrefs

Cf. A000041, A006128, A135010, A138121, A186114, A186412, A193870, A194437, A194438, A194439, A194446, A194447.

A194439 Number of regions in the set of partitions of n that contain only one part.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297
Offset: 1

Views

Author

Omar E. Pol, Nov 28 2011

Keywords

Comments

It appears that this is 1 together with A000041. - Omar E. Pol, Nov 29 2011
For the definition of "region" see A206437. See also A186114 and A193870.

Examples

			For n = 5 the seven regions of 5 in nondecreasing order are the sets of positive integers of the rows as shown below:
   1;
   1, 2;
   1, 1, 3;
   0, 0, 0, 2;
   1, 1, 1, 2, 4;
   0, 0, 0, 0, 0, 3;
   1, 1, 1, 1, 1, 2, 5;
   ...
There are three regions that contain only one positive part, so a(5) = 3.
Note that in every column of the triangle the positive integers are also the parts of one of the partitions of 5.

Links

Crossrefs

Column 1 of A194438.
Cf. A000041, A002865, A027336, A135010, A138121, A186114, A186412, A193870, A194436, A194437, A194446, A194447, A206437.

Formula

It appears that a(n) = A000041(n-2), if n >= 2. - Omar E. Pol, Nov 29 2011
It appears that a(n) = A000041(n) - A027336(n), if n >= 2. - Omar E. Pol, Nov 30 2011

Extensions

Definition clarified by Omar E. Pol, May 21 2021

A207779 Largest part plus the number of parts of the n-th region of the section model of partitions.

Original entry on oeis.org

2, 4, 6, 3, 9, 4, 12, 3, 6, 4, 17, 4, 7, 5, 22, 3, 6, 4, 10, 6, 5, 30, 4, 7, 5, 11, 4, 8, 6, 39, 3, 6, 4, 10, 6, 5, 15, 5, 9, 7, 6, 52, 4, 7, 5, 11, 4, 8, 6, 17, 6, 5, 11, 8, 7, 67, 3, 6, 4, 10, 6, 5, 15, 5, 9, 7, 6, 22, 4, 8, 6, 13, 5, 10, 8
Offset: 1

Views

Author

Omar E. Pol, Mar 08 2012

Keywords

Comments

Also semiperimeter of the n-th region of the geometric version of the section model of partitions. Note that a(n) is easily viewable as the sum of two perpendicular segments with a shared vertex. The horizontal segment has length A141285(n) and the vertical segment has length A194446(n). The difference between these two segments gives A194447(n). See also an illustration in the Links section. For the definition of "region" see A206437.
Also triangle read by rows: T(n,k) = largest part plus the number of parts of the k-th region of the last section of the set of partitions of n.

Examples

			Written as a triangle begins:
2;
4;
6;
3, 9;
4, 12;
3, 6, 4, 17;
4, 7, 5, 22;
3, 6, 4, 10, 6, 5, 30;
4, 7, 5, 11, 4, 8, 6, 39;
3, 6, 4, 10, 6, 5, 15, 5, 9, 7, 6, 52;

Links

Crossrefs

Row n has length A187219(n). Last term of row n is A133041(n). Where record occur give A000041, n >= 1.
Cf. A002865, A135010, A182699, A182709, A183152, A194436, A194437, A194438, A194439, A194447, A206437.

Formula

a(n) = A141285(n) + A194446(n).

A194438 Triangle read by rows: T(n,k) is the number of regions of the set of partitions of n into k parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 3, 1, 1, 0, 1, 0, 1, 5, 2, 1, 0, 1, 0, 1, 0, 0, 0, 1, 7, 3, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 11, 4, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 15, 6, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0
Offset: 1

Views

Author

Omar E. Pol, Nov 28 2011

Keywords

Comments

For the definition of "region" see A206437. See also A186114 and A193870. - Omar E. Pol, May 21 2021

Examples

			Triangle begins:
   1;
   1,1;
   1,1,1;
   2,1,1,0,1;
   3,1,1,0,1,0,1;
   5,2,1,0,1,0,1,0,0,0,1;
   7,3,1,0,1,0,1,0,0,0,1,0,0,0,1;
  11,4,1,1,1,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1;
...

Links

Crossrefs

Column 1 is A194439.
Row n has length A000041(n).
Row sums give A000041, n >= 1.
Cf. A008284, A135010, A138121, A186114, A186412, A193870, A194436, A194437, A194439, A194446, A194447, A206437.

Extensions

Definition clarified by Omar E. Pol, May 21 2021

A194448 Number of parts > 1 in the n-th region of the shell model of partitions.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 3, 1, 1, 14, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 1, 21, 1, 2, 1, 4, 1, 2
Offset: 1

Views

Author

Omar E. Pol, Dec 10 2011

Keywords

Comments

Also triangle read by rows: T(n,k) = number of parts > 1 in the k-th region of the last section of the set of partitions of n.

Examples

			Written as a triangle:
0;
1;
1;
1,2;
1,2;
1,2,1,4;
1,2,1,4;
1,2,1,4,1,1,7;
1,2,1,4,1,2,1,8;
1,2,1,4,1,1,7,1,2,1,1,12;
1,2,1,4,1,2,1,8,1,1,3,1,1,14;
1,2,1,4,1,1,7,1,2,1,1,12,1,2,1,4,1,2,1,1,21;

Crossrefs

Row sums give A138135. Row n has length A187219(n).
Cf. A000041, A002865, A135010, A138121, A138137, A138879, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A194447, A194449.

A194449 Largest part minus the number of parts > 1 in the n-th region of the set of partitions of j, if 1 <= n <= A000041(j).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 2, 2, 3, 3, 3, 1, 2, 2, 2, 4, 3, 1, 2, 3, 3, 3, 2, 4, 4, 1, 1, 2, 2, 2, 4, 3, 1, 3, 5, 5, 4, -2, 2, 3, 3, 3, 2, 4, 4, 1, 4, 3, 5, 6, 5, -3, 1, 2, 2, 2, 4, 3, 1, 3, 5, 5, 4, -2, 2, 4, 4, 5, 3, 6, 6, 5, -9
Offset: 1

Views

Author

Omar E. Pol, Dec 10 2011

Keywords

Comments

Also triangle read by rows: T(j,k) = largest part minus the numbers of parts > 1 in the k-th region of the last section of the set of partitions of j. It appears that the sum of row j is equal to A000041(j-1). For the definition of "region" of the set of partitions of j see A206437. See also A135010.

Examples

			The 7th region of the shell model of partitions is [5, 2, 1, 1, 1, 1, 1]. The largest part is 5 and the number of parts > 1 is 2, so a(7) = 5 - 2 = 3 (see an illustration in the link section).
Written as an irregular triangle T(j,k) begins:
1;
1;
2;
1,2;
2,3;
1,2,2,2;
2,3,3,3;
1,2,2,2,4,3,1;
2,3,3,3,2,4,4,1;
1,2,2,2,4,3,1,3,5,5,4,-2;
2,3,3,3,2,4,4,1,4,3,5,6,5,-3;
1,2,2,2,4,3,1,3,5,5,4,-2,2,4,4,5,3,6,6,5,-9;

Links

Crossrefs

Row j has length A187219(j).
Cf. A000041, A135010, A138121, A138137, A138879, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A194447, A206437.

Formula

a(n) = A141285(n) - A194448(n).

A206441 Triangle read by rows. T(n,k) = number of distinct parts in the k-th region of the last section of the set of partitions of n.

Original entry on oeis.org

1, 2, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 3, 1, 1, 5, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 1, 6
Offset: 1

Views

Author

Omar E. Pol, Feb 13 2012

Keywords

Comments

a(n) is also the number of distinct parts in the n-th region of the shell model of partitions (see A135010 and A206437).

Examples

			The first region in the last section of the set of partitions of 6 looks like this:
.        **
There is only one part, so T(6,1) = 1.
The second region in the last section of the set of partitions of 6 looks like this:
.        ****
.          **
There are two distinct parts, so T(6,2) = 2.
The third region in the last section of the set of partitions of 6 looks like this:
.        ***
There is only one part, so T(6,3) = 1.
The 4th region in the last section of the set of partitions of 6 looks like this:
.        ******
.           ***
.            **
.            **
.             *
.             *
.             *
.             *
.             *
.             *
.             *
There are four distinct parts, so T(6,4) = 4.
Written as a triangle:
1;
2;
2;
1, 3;
1, 3;
1, 2, 1, 4;
1, 2, 1, 4;
1, 2, 1, 3, 1, 1, 5;
1, 2, 1, 3, 1, 2, 1, 5;
1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 1, 6;

Crossrefs

Row n has length A187219(n).
Cf. A135010, A138121, A141285, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A194447.
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.